Semiclassical Nodal Solutions for the Choquard equation

Abstract

In this paper, we are going to investigate the existence of semiclassical nodal solutions for the following Choquard equation: \begin{equation}\nonumber -\varepsilon^{2}\Delta u+V(x)u= (I_{\alpha} \ast|u|^p)|u|^{p-2}u, \ \ \ \ \text{in}\quad \mathbb R^N, \end{equation} where $N\ge 3$, $\alpha\in (0,N),$ and $I_{\alpha}$ is the Riesz potential. Under suitable assumption on the potential $V(x)$, for $p\in (2,\frac{N+\alpha}{N-2})$, the semiclassical nodal solutions are obtained if the parameter $\varepsilon$ is small enough. Meanwhile, we also prove that there is no least energy nodal solutions if $p < 2$. Moreover, the limit of the least energy nodal solutions of the equation is used to solve the case of $p=2$.